Optimal. Leaf size=35 \[ \frac {(a+b x) \log (a+b x)}{b \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {608, 31} \[ \frac {(a+b x) \log (a+b x)}{b \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 608
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(a+b x) \log (a+b x)}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.74 \[ \frac {(a+b x) \log (a+b x)}{b \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 10, normalized size = 0.29 \[ \frac {\log \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 17, normalized size = 0.49 \[ \frac {\log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 0.71 \[ \frac {\left (b x +a \right ) \ln \left (b x +a \right )}{\sqrt {\left (b x +a \right )^{2}}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 12, normalized size = 0.34 \[ \frac {\log \left (x + \frac {a}{b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 19, normalized size = 0.54 \[ \frac {\ln \left (a+b\,x+\sqrt {{\left (a+b\,x\right )}^2}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 7, normalized size = 0.20 \[ \frac {\log {\left (a + b x \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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